# How to prove √X is irrational number

by SOHAWONG
Tags: √x, irrational, number, prove
 P: 16 when X is even number,it's easy to prove but how about the condition which X is odd number? I have no idea of this
 Emeritus Sci Advisor PF Gold P: 16,091 $\sqrt{4}$ is irrational?
P: 16
 Quote by Hurkyl $\sqrt{4}$ is irrational?

 PF Gold P: 1,956 How to prove √X is irrational number So in other words... $$\sqrt{x}$$ is irrational iff x=/=n^2 for n belonging to the integer set.
P: 16
 Quote by Char. Limit So in other words... $$\sqrt{x}$$ is irrational iff x=/=n^2 for n belonging to the integer set.
yes, but how to prove?
 P: 234 Fundamental theorem of arithmetic. Assume p^2/q^2=x with gcd(p,q)=1, and see what has to divide what.
P: 16
 Quote by Tinyboss Fundamental theorem of arithmetic. Assume p^2/q^2=x with gcd(p,q)=1, and see what has to divide what.
what does gcd mean?
 P: 234 Greatest common divisor. If gcd(p,q)=1, it means the fraction p/q is in lowest terms. Look at the proof for sqrt(2), and adapt it. Remember that "even" just means "is divisible by 2", so that if you're checking a number other than 2, you won't be thinking about "even" anymore.
 P: 1,822 Although right now you're probably more interested in just getting the right answer, you might want to check out the following Wikipedia entries: http://en.wikipedia.org/wiki/Square_root http://en.wikipedia.org/wiki/Square_root_of_2 And for an interesting history of the discovery of irrational numbers look at http://en.wikipedia.org/wiki/Irrational_number

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